Use ConvNet to predict bitmap
Your first approach doesn't make any sense to me. After all, you are not just interested in the centre pixel are you? And if you have two nodes for every pixel, what are those two nodes encoding? For the probability of water you just need one. 

So clearly approach two. I would just use 0.5 as cutoff. Using a higher or lower cutoff only makes sense, if either false positives or false negatives are for some reason more problematic. Using some additional heuristic to adjust the given probabilities would just do what the ConvNet should already have done.

If 4096 output nodes is too expensive you can always make it a bit fuzzier, for example by predicting the probability that at least one of four pixels shows water, reducing the number of nodes to 1024.